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CS 267 Applications of Parallel Computers Lecture 15: Graph Partitioning - II

CS 267 Applications of Parallel Computers Lecture 15: Graph Partitioning - II. James Demmel http://www.cs.berkeley.edu/~demmel/cs267_Spr99. Outline of Graph Partitioning Lectures. Review of last lecture Partitioning without Nodal Coordinates - continued Kernighan/Lin Spectral Partitioning

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CS 267 Applications of Parallel Computers Lecture 15: Graph Partitioning - II

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  1. CS 267 Applications of Parallel ComputersLecture 15: Graph Partitioning - II James Demmel http://www.cs.berkeley.edu/~demmel/cs267_Spr99

  2. Outline of Graph Partitioning Lectures • Review of last lecture • Partitioning without Nodal Coordinates - continued • Kernighan/Lin • Spectral Partitioning • Multilevel Acceleration • BIG IDEA, will appear often in course • Available Software • good sequential and parallel software availble • Comparison of Methods • Applications

  3. Review Definition of Graph Partitioning • Given a graph G = (N, E, WN, WE) • N = nodes (or vertices), E = edges • WN = node weights,WE = edge weights • Ex: N = {tasks}, WN = {task costs}, edge (j,k) in E means task j sends WE(j,k) words to task k • Choose a partition N = N1 U N2 U … U NP such that • The sum of the node weights in each Nj is “about the same” • The sum of all edge weights of edges connecting all different pairs Nj and Nk is minimized • Ex: balance the work load, while minimizing communication • Special case of N = N1 U N2: Graph Bisection

  4. Review of last lecture • Partitioning with nodal coordinates • Rely on graphs having nodes connected (mostly) to “nearest neighbors” in space • Common when graph arises from physical model • Algorithm very efficient, does not depend on edges! • Can be used as good starting guess for subsequent partitioners, which do examine edges • Can do poorly if graph less connected: • Partitioning without nodal coordinates • Depends on edges • No assumptions about where “nearest neighbors” are • Began with Breadth First Search (BFS)

  5. Partitioning without nodal coordinates - Kernighan/Lin • Take a initial partition and iteratively improve it • Kernighan/Lin (1970), cost = O(|N|3) but easy to understand • Fiduccia/Mattheyses (1982), cost = O(|E|), much better, but more complicated • Let G = (N,E,WE) be partitioned as N = A U B, where |A| = |B| • T = cost(A,B) = S {W(e) where e connects nodes in A and B} • Find subsets X of A and Y of B with |X| = |Y| so that swapping X and Y decreases cost: • newA = A - X U Y and newB = B - Y U X • newT = cost(newA , newB) < cost(A,B) • Keep choosing X and Y until cost no longer decreases • Need to compute newT efficiently for many possible X and Y, choose smallest

  6. Kernighan/Lin - Preliminary Definitions • T = cost(A, B), newT = cost(newA, newB) • Need an efficient formula for newT; will use • E(a) = external cost of a in A = S {W(a,b) for b in B} • I(a) = internal cost of a in A = S {W(a,a’) for other a’ in A} • D(a) = cost of a in A = E(a) - I(a) • Moving a from A to B would decrease T by D(a) • E(b), I(b) and D(b) defined analogously for b in B • Consider swapping X = {a} and Y = {b} • newA = A - {a} U {b}, newB = B - {b} U {a} • newT = T - ( D(a) + D(b) - 2*w(a,b) ) = T - gain(a,b) • gain(a,b) measures improvement gotten by swapping a and b • Update formulas, after a and b are swapped • newD(a’) = D(a’) + 2*w(a’,a) - 2*w(a’,b) for a’ in A, a’ != a • newD(b’) = D(b’) + 2*w(b’,b) - 2*w(b’,a) for b’ in B, b’ != b

  7. Kernighan/Lin Algorithm Compute T = cost(A,B) for initial A, B … cost = O(|N|2) Repeat … One pass greedily computes |N|/2 possible X,Y to swap, picks best Compute costs D(n) for all n in N … cost = O(|N|2) Unmark all nodes in N … cost = O(|N|) While there are unmarked nodes … |N|/2 iterations Find an unmarked pair (a,b) maximizing gain(a,b) … cost = O(|N|2) Mark a and b (but do not swap them) … cost = O(1) Update D(n) for all unmarked n, as though a and b had been swapped … cost = O(|N|) Endwhile … At this point we have computed a sequence of pairs … (a1,b1), … , (ak,bk) and gains gain(1),…., gain(k) … where k = |N|/2, numbered in the order in which we marked them Pick m maximizing Gain = Sk=1 to m gain(k) … cost = O(|N|) … Gain is reduction in cost from swapping (a1,b1) through (am,bm) If Gain > 0 then … it is worth swapping Update newA = A - { a1,…,am } U { b1,…,bm } … cost = O(|N|) Update newB = B - { b1,…,bm } U { a1,…,am } … cost = O(|N|) Update T = T - Gain … cost = O(1) endif Until Gain <= 0

  8. Comments on Kernighan/Lin Algorithm • Most expensive line show in red • Some gain(k) may be negative, but if later gains are large, then final Gain may be positive • can escape “local minima” where switching no pair helps • How many times do we Repeat? • K/L tested on very small graphs (|N|<=360) and got convergence after 2-4 sweeps • For random graphs (of theoretical interest) the probability of convergence in one step appears to drop like 2-|N|/30

  9. Partitioning without nodal coordinates - Spectral Bisection • Based on theory of Fiedler (1970s), popularized by Pothen, Simon, Liou (1990) • Motivation, by analogy to a vibrating string • Basic definitions • Vibrating string, revisited • Motivation, by using a continuous approximation to a discrete optimization problem • Implementation via the Lanczos Algorithm • To optimize sparse-matrix-vector multiply, we graph partition • To graph partition, we find an eigenvector of a matrix associated with the graph • To find an eigenvector, we do sparse-matrix vector multiply • No free lunch ...

  10. Motivation for Spectral Bisection: Vibrating String • Think of G = 1D mesh as masses (nodes) connected by springs (edges), i.e. a string that can vibrate • Vibrating string has modes of vibration, or harmonics • Label nodes by whether mode - or + to partition into N- and N+ • Same idea for other graphs (eg planar graph ~ trampoline)

  11. Basic Definitions • Definition: The incidence matrix In(G) of a graph G(N,E) is an |N| by |E| matrix, with one row for each node and one column for each edge. If edge e=(i,j) then column e of In(G) is zero except for the i-th and j-th entries, which are +1 and -1, respectively. • Slightly ambiguous definition because multiplying column e of In(G) by -1 still satisfies the definition, but this won’t matter... • Definition: The Laplacian matrix L(G) of a graph G(N,E) is an |N| by |N| symmetric matrix, with one row and column for each node. It is defined by • L(G) (i,i) = degree of node I (number of incident edges) • L(G) (i,j) = -1 if i != j and there is an edge (i,j) • L(G) (i,j) = 0 otherwise

  12. Example of In(G) and L(G) for 1D and 2D meshes

  13. Properties of Incidence and Laplacian matrices • Theorem 1: Given G, In(G) and L(G) have the following properties (proof on web page) • L(G) is symmetric. (This means the eigenvalues of L(G) are real and its eigenvectors are real and orthogonal.) • Let e = [1,…,1]T, i.e. the column vector of all ones. Then L(G)*e=0. • In(G) * (In(G))T = L(G). This is independent of the signs chosen for each column of In(G). • Suppose L(G)*v = l*v, v != 0, so that v is an eigenvector and l an eigenvalue of L(G). Then • The eigenvalues of L(G) are nonnegative: • 0 = l1 <= l2 <= … <= ln • The number of connected components of G is equal to the number of li equal to 0. In particular, l2 != 0 if and only if G is connected. • Definition: l2(L(G)) is the algebraic connectivity of G l = || In(G)T * v ||2 / || v ||2 … ||x||2 = Sk xk2 = S { (v(i)-v(j))2 for all edges e=(i,j) } / Si v(i)2

  14. Spectral Bisection Algorithm • Spectral Bisection Algorithm: • Compute eigenvector v2 corresponding to l2(L(G)) • For each node n of G • if v2(n) < 0 put node n in partition N- • else put node n in partition N+ • Why does this make sense? First reasons... • Theorem 2 (Fiedler, 1975): Let G be connected, and N- and N+ defined as above. Then N- is connected. If no v2(n) = 0, then N+ is also connected. (proof on web page) • Recall l2(L(G)) is the algebraic connectivity of G • Theorem 3 (Fiedler): Let G1(N,E1) be a subgraph of G(N,E), so that G1 is “less connected” than G. Then l2(L(G)) <= l2(L(G)) , i.e. the algebraic connectivity of G1 is less than or equal to the algebraic connectivity of G. (proof on web page)

  15. Motivation for Spectral Bisection: Vibrating String • Vibrating string has modes of vibration, or harmonics • Modes computable as follows • Model string as masses connected by springs (a 1D mesh) • Write down F=ma for coupled system, get matrix A • Eigenvalues and eigenvectors of A are frequencies and shapes of modes • Label nodes by whether mode - or + to get N- and N+ • Same idea for other graphs (eg planar graph ~ trampoline)

  16. Details for vibrating string • Force on mass j = k*[x(j-1) - x(j)] + k*[x(j+1) - x(j)] = -k*[-x(j-1) + 2*x(j) - x(j+1)] • F=ma yields m*x’’(j) = -k*[-x(j-1) + 2*x(j) - x(j+1)] (*) • Writing (*) for j=1,2,…,n yields x(1) 2*x(1) - x(2) 2 -1 x(1) x(1) x(2) -x(1) + 2*x(2) - x(3) -1 2 -1 x(2) x(2) m * d2 … =-k* … =-k* … * … =-k*L* … dx2 x(j) -x(j-1) + 2*x(j) - x(j+1) -1 2 -1 x(j) x(j) … … … … … x(n) 2*x(n-1) - x(n) -1 2 x(n) x(n) (-m/k) x’’ = L*x

  17. Details for vibrating string - continued • -(m/k) x’’ = L*x, where x = [x1,x2,…,xn ]T • Seek solution of form x(t) = sin(a*t) * x0 • L*x0 = (m/k)*a2 * x0 = l * x0 • For each integer i, get l = 2*(1-cos(i*p/(n+1)), x0 = sin(1*i*p/(n+1)) sin(2*i*p/(n+1)) … sin(n*i*p/(n+1)) • Thus x0 is a sine curve with frequency proportional to i • Thus a2 = 2*k/m *(1-cos(i*p/(n+1)) or a ~ sqrt(k/m)*p*i/(n+1) • L = 2 -1 not quite L(1D mesh), -1 2 -1 but we can fix that ... …. -1 2

  18. A “vibrating string” for L(1D mesh) • First equation changes to m*x’’(1) = -k*[-x(2)+ 2x(1)] • First row of T changes from [ 2 -1 0 … ] to [ 1 -1 0 … ] • Last equation changes to m*x’’(n)=-k*[-x(n-1) + 2x(n)] • Last row of T changes from [ … 0 -1 2 ] to [ … 0 -1 1 ] • Component j of i-th eigenvector changes to cos((j-.5)*(i-1)*p/n)

  19. Eigenvectors of L(1D mesh) Eigenvector 1 (all ones) Eigenvector 2 Eigenvector 3

  20. 2nd eigenvector of L(planar mesh)

  21. 4th eigenvector of L(planar mesh)

  22. Motivation for Spectral Bisection:Continuous Approximation to a discrete optimization problem • Use L(G) to count the number of edges from N- to N+ • Lemma 1: Let N = N- U N+ be a partition of G(N,E). Let x(j) = -1 if j is in N- and x(j) = +1 if j is in N+. Then (proof on web page) • Restate partitioning problem as finding vector x with entries +1 or -1 such that • Sk x(k) = 0, i.e. |N+| = |N-| • # edges connecting N+ to N- = .25*xT*L(G)*x is minimized • Put node j in N+ (or N-) if x(j) >=0 (or < 0) The number of edges connecting N- and N+ = .25 * xT * L(G) * x = .25 * Si,k x(i) * L(G)(i,k) * x(k) = .25 * S { (x(i) - x(k))2 for all edges (i,j) }

  23. Converting a discrete to a continuous problem • Discrete: Find x with entries +1 or -1 such that • Sk x(k) = 0, i.e. |N+| = |N-| • # edges connecting N+ to N- = .25*xT*L(G)*x is minimized • Put node j in N+ (or N-) if x(j) >=0 (or < 0) • Continuous: Find x with real entries such that • Sk x(k) = 0 and Sk (x(k))2 = |N| (set includes discrete one above) • .25*xT*L(G)*x is minimized • Put node j in N+ (or N-) if x(j) >=0 (or < 0) • Theorem 4 (Courant/Fischer “minimax theorem”): x satisfying continuous problem is eigenvector v2, for l2 . (proof on web page) • Theorem 5: Theminimum number of edges connecting N+ and N- in any partitioning with |N+|=|N-| is at least .25*|N|* l2. (proof on web page) • The larger the algebraic connectivityl2, the more edges we need to cut to bisect the graph

  24. Computing v2 and l2 of L(G) using Lanczos • Given any n-by-n symmetric matrix A(such as L(G))Lanczos computes a k-by-k “approximation” T by doing k matrix-vector products, k << n • Approximate A’s eigenvalues/vectors using T’s Choose an arbitrary starting vector r b(0) = ||r|| j=0 repeat j=j+1 q(j) = r/b(j-1) … scale a vector r = A*q(j) … matrix vector multiplication, the most expensive step r = r - b(j-1)*v(j-1) … “saxpy”, or scalar*vector + vector a(j) = v(j)T * r … dot product r = r - a(j)*v(j) … “saxpy” b(j) = ||r|| … compute vector norm until convergence … details omitted T = a(1) b(1) b(1) a(2) b(2) b(2) a(3) b(3) … … … b(k-2) a(k-1) b(k-1) b(k-1) a(k)

  25. References • Details of all proofs on web page • A. Pothen, H. Simon, K.-P. Liou, “Partitioning sparse matrices with eigenvectors of graphs”, SIAM J. Mat. Anal. Appl. 11:430-452 (1990) • M. Fiedler, “Algebraic Connectivity of Graphs”, Czech. Math. J., 23:298-305 (1973) • M. Fiedler, Czech. Math. J., 25:619-637 (1975) • B. Parlett, “The Symmetric Eigenproblem”, Prentice-Hall, 1980 • www.cs.berkeley.edu/~ruhe/lantplht/lantplht.html • www.netlib.org/laso

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